Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470374 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,7,8,6)(9,10,13,15)(11,14,17,21)(12,19,20,16)(18,24,23,22)(25,26)(27,36)(28,33)(29,34)(30,44)(31,35)(32,39)(37,50)(38,46)(40,51)(41,47)(42,48)(43,49)(45,52)(53,55)(54,56);(2,5)(6,7)(9,11,13,17)(10,14,15,21)(12,18,20,23)(16,22,19,24)(25,27,30,39,31,40,29,38)(26,32,34,36,35,46,44,51)(28,37,41,52,42,53,43,54)(33,45,49,50,48,56,47,55);(1,3,4,8)(2,6,5,7)(10,15)(11,17)(16,19)(18,23)(25,28)(26,33)(27,37)(29,43)(30,41)(31,42)(32,45)(34,49)(35,48)(36,50)(38,54)(39,52)(40,53)(44,47)(46,56)(51,55);(11,17)(14,21)(18,23)(22,24)(25,29,31,30)(26,34,35,44)(27,38,40,39)(28,41,42,43)(32,36,46,51)(33,47,48,49)(37,52,53,54)(45,55,56,50);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(28,42)(33,48)(37,53)(41,43)(45,56)(47,49)(50,55)(52,54)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=gh,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406679);